My understanding is that forcing (such as Cohen forcing) can be described via a topos.  For example this [nlab article on forcing][1] describes forcing as a "the topos of sheaves on a suitable site."  

My question concerns forcing in computability theory, for example as described in Chapter 3 or these [lecture notes of Richard Shore][2].  The idea is that the generics are those which meet all *computable* dense sets of forcing conditions.  (Computable can mean a few things.  Often it is taken to mean a $\Sigma^0_1$ set of forcing conditions.  Also, usually the forcing posets are countable.)  Since there are only countably many such dense sets, such effective generics exist.

>Is there a known/canonical type of topos corresponding to the forcing in computability theory?

Any references would be appreciated.

*FYI: My background is in computability theory, proof theory, and computable analysis.  I know little about topos theory, but I am willing to learn a bit.  I am mostly asking this question because I want to compare some ideas I have about effective versions of Solovay forcing with some work by others about the topos corresponding to Solovay forcing.  Also, it is always nice to learn new things.*

  [1]: http://ncatlab.org/nlab/show/forcing
  [2]: http://www.math.cornell.edu/~shore/papers/pdf/SingLect2NS.pdf